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# On the evolution of harming and recognition in finite panmictic and infinite structured populations

^{*}Department of Biology, Stanford University, USA

^{†}Department of Biology, Stanford University, USA

^{‡}Laboratoire Génétique et Environnement, Université de Montpellier II, France

## Abstract

Natural selection may favor two very different types of social behaviors that have costs in vital rates (fecundity and/or survival) to the actor: helping behaviors, which increase the vital rates of recipients, and harming behaviors, which reduce the vital rates of recipients. While social evolutionary theory has mainly dealt with helping behaviors, competition for limited resources creates ecological conditions where an actor may benefit from expressing behaviors that reduce the vital rates of neighbours. This may occur if the reduction in vital rates decreases the intensity of competition experienced by the actor or that experienced by its offspring. Here, we explore the joint evolution of neutral recognition markers and marker-based costly conditional harming whereby actors express harming, conditional on actor and recipient bearing different conspicuous markers. We do so for two complementary demographic scenarios: finite panmictic and infinite structured populations. We find that marker-based conditional harming can evolve under a large range of recombination rates and group sizes under both finite panmictic and infinite structured populations. Direct comparison with results for the evolution of marker-based conditional helping reveals that, if everything else is equal, marker-based conditional harming is often more likely to evolve than marker-based conditional.

**Keywords:**harming, helping, spite, marker recognition, competition, genetic relationship, genetic diversity, fixation probabilities, two-locus models

## Introduction

Helping behaviors, by which individuals in a population provide fitness benefits to others, are more likely to evolve in the presence of mechanisms that allow discrimination against defectors (Hamilton, 1964, 1971; Axelrod and Hamilton, 1981; Eshel and Cavalli-Sforza, 1982). Discrimination may occur if helpers can recognize each other by using conspicuous phenotypic cues, tags, or markers, and provide benefits to other individuals carrying the gene(s) underlying helping, instead of providing benefits to defectors. Both genetic kin-recognition and the so-called green-beard mechanism may involve the expression of helping conditional based on actor and recipient bearing identical recognition markers (Hamilton, 1964; Dawkins, 1982; Grafen, 1990). Such conditional helping relies on a tight linkage between the genes underlying helping and those producing the conspicuous markers. If genes for helping and markers were loosely coupled, defectors could acquire the markers expressed by helpers and then receive the benefits of helping without paying the cost. This would ultimately prevent the evolution of marker-based conditional helping.

Genetic kin-recognition based on actor and recipient bearing identical marker alleles might evolve in spatially-structured populations (Axelrod et al., 2004; Jansen and van Baalen, 2006; Rousset and Roze, 2007). Two individuals from the same group (or spatial location) are more likely to have inherited both helping and recognition marker alleles from the same recent common ancestor than are two individuals from different groups. Common ancestry can then lead to the build up of genetic associations between helping and recognition alleles between individuals, which descend from the same group even in the presence of recombination, provided migration is limited and group size is not too large. Because individuals within groups might interact with resident and immigrant individuals, recognition markers that are identical-by-descent may allow actors to discriminate among categories of recipients defined by markers within groups. Such marker-based discrimination then sustains the evolution of conditional helping under strong population structure (Axelrod et al., 2004; Jansen and van Baalen, 2006; Rousset and Roze, 2007).

An increase in group size or migration rate erodes population structure and weakens the genetic associations between individuals from the same group, within and across loci. With frequent migration, population structure is likely to vanish. In this case, marker-based helping is no longer expected to evolve. Nevertheless, if the population becomes panmictic but is of finite size, some variation may remain in the propensity of interacting individuals to share alleles identical-by-descent at many loci. Indeed, two offspring of the same parent are always more likely to have inherited identical helping and recognition marker alleles than are two individuals sampled at random from the population. The variation in the ancestry of pairs of interacting individuals within a panmictic population might then still allow actors to discriminate among categories of recipients. This variance supports the evolution of marker-based conditional helping when population size is very small (Traulsen and Nowak, 2007) because the probability that individuals descending from the same parent interact in a panmictic population is likely to be small, approximately equal to the inverse of population size in the absence of searching. Thus, whether marker-based conditional helping evolves in panmictic or structured populations, finite population size is a crucial demographic requirement for the evolution of the behavior.

With finite local group (or total population) size individuals that help each other are also more likely to compete for the same local (or global) resources. Competition between interacting individuals has been shown to partially offset the benefits of helping in both finite panmictic (Hamilton, 1971) and infinite structured populations (Taylor, 1992a,b). Finite population size thus creates ecological conditions where actors may actually benefit from expressing behaviors that reduce the fecundity of neighbours by harming them, instead of increasing it by helping them, even when the actor suffers a fecundity cost. This follows from the fact that in a finite population a single individual may have a marked effect on population productivity and decrease the intensity of competition experienced by its offspring.

From the green-beard in the red fire ant *Solenopsis invicta*, which kills individuals that do not have it, to bacteria releasing antagonistic compounds in their environment, to maternally transmitted symbionts generating cytoplasmic incompatibility, several examples have been documented where genotypes spread through natural populations by hampering the reproduction of those that do not carry them, thereby reducing the intensity of competition experienced by their carriers (e.g., Werren, 1997; Keller and Ross, 1998; Riley and Gordon, 1999; Brown et al., 2006). It is thus useful to understand not only what are the ecological and demographic conditions leading to the evolution of helping behaviors (cooperation and altruism), but also those conducive to the evolution of harming (exploitation and spite). This might lead to a better understanding of the type of social interactions expected to occur in natural populations.

In this paper, we try to understand the conditions under which marker-based conditional harming, whereby an actor decreases the fecundity of recipients conditional on them bearing a different phenotypic cue than the actor, is selected for. To that aim, we analyze the joint evolution of neutral recognition markers and marker-based conditional harming behaviors in a two-locus population genetic framework. The first locus controls the expression of neutral conspicuous markers. The second locus determines the expression of harming, conditional on actor and recipient bearing different conspicuous markers at the first locus. We show that under a Wright-Fisher scheme of reproduction, marker-based conditional harming can evolve for a large range of recombination rates and group sizes, in both finite panmictic and infinite structured populations. Direct comparison with results for the evolution of marker-based conditional helping reveals conditions under which, everything else being equal, the selective pressure favoring marker-based conditional harming is stronger than that on conditional helping. In particular, this is the case when only two conspicuous marker alleles at the recognition locus segregate in the population.

## Model

### Life-cycle

We consider a population with an infinite number of groups each of finite size *N* (a list of symbols is given in Table 1). Individuals within groups interact with each of their *N* − 1 neighbours. We assume that individuals are haploid and carry two different loci, with two alleles segregating at each locus. The first locus determines the expression of a conspicuous phenotypic feature, which is the basis of recognition: individuals bearing a mutant recognition allele, R, at this locus express a different phenotype from those bearing the wild-type, resident allele, r. Both mutant and resident alleles have the same effect on reproduction and can therefore be regarded as neutral alleles. The second locus controls the expression of a harming behavior. A focal actor bearing a mutant harming allele H at this locus decreases the fecundity of a single recipient by *D*/(*N* − 1) at a direct fecundity cost *C*/(*N* − 1) to itself, conditional on the recipient bearing a different allele at the recognition locus than the actor. Those individuals that carry the resident allele, h, at the harming locus do not express any phenotype, but their fecundity is decreased whenever they interact with individuals carrying the harming allele and a different neutral conspicuous marker.

These phenotypic effects on fitness entail that if harming were expressed unconditionally, regardless of the recipient’s genotype, then the act of harming would result in a net fecundity cost *C* to the actor [(*N* − 1) acts times the cost *C*/(*N* − 1) per act] and a total effect *D* on group productivity [(*N* − 1) acts times the cost *D*/(*N* − 1) to each recipient]. This corresponds to the usual parameterization of the evolution of unconditional helping in patch structured populations, with the difference that the effect *D* on group productivity is a benefit in that case (e.g., Eshel, 1972; Aoki, 1982; Taylor, 1992a; Gardner and West, 2006). The conditional expression of the behavior will result in a situation where the cost to the actor (or the total effect on group productivity) varies between zero (never expressing the act) and *C* (or *D*).

Events in the life-cycle of the population occur in the following order. (1) Each adult individual produces a large number of juveniles with its fecundity being determined by the interactions with its *N* − 1 neighbours. (2) Juveniles disperse independently from each other with probability *m* to another random group, and all adults die. (3) Juveniles fuse randomly to produce diploid zygotes (syngamy), which is immediately followed by meiosis with a recombination rate *r* between the harming and recognition loci to produce a new generation of haploid individuals. (4) Regulation occurs so that only *N* individuals are allowed to settle in each group.

### Gene frequency change and fixation probability

We denote by *p*_{A(}_{ij}_{)} the frequency (0 or 1) of a mutant allele A (A stands for H or R) in individual *j* from group *i*. Individuals *j* and *k* from group *i* then bear different marker alleles with probability *p*_{R(}_{ij}_{)}(1 − *p*_{R(}_{ik}_{)}) + (1 − *p*_{R(}_{ij}_{)})*p*_{R(}_{ik}_{)}. With our assumptions that individuals interact with each of their *N* − 1 group neighbours and that actors harm only recipients bearing different marker alleles (and are harmed by others bearing the harming allele and different maker alleles), the fecundity of individual *j* from group *i* (relative to that of an individual not expressing the harming allele) can be written as

The expected change in the frequency *p*_{A} of allele A over the life cycle, conditional on the current distribution of genotypes in the population (i.e., the current values of *p*_{H(}_{ij}_{)} and *p*_{R(}_{ij}_{)} for each individual in the population), is then given to leading order in *C* and *D* (weak selection) by

where E_{i}_{,}* _{j}*[·] denotes the average over all groups in the population and all individuals within groups (

*p*

_{A}= E

_{i}_{,}

*[*

_{j}*p*

_{A(}

_{ij}_{)}]); 1 +

*f*is the average fecundity in group

_{i}*i*(average over 1 +

*f*within group

_{ij}*i*); and 1+

*f*is the average fecundity in the total population (eqs. A-1 and A-2 of the Appendix).

On substitution of eq. 1 into eq. 2 one obtains an expression for the expected change in allele frequency Δ*p*_{A}, which is a sum of selection coefficients (e.g. *C*, *D*, (1 − *m*)^{2}*C*, etc.), each weighted by moments of allele frequencies involving genes from the same or different loci, sampled from the same or different individuals (e.g. E_{i}_{,}_{j}_{,}_{k}_{≠}* _{j}*[

*p*

_{H(}

_{ij}_{)}

*p*

_{H(}

_{ik}_{)}], E

_{i}_{,}

_{j}_{,}

_{k}_{≠}

*[*

_{j}*p*

_{H(}

_{ij}_{)}

*p*

_{R(}

_{ik}_{)}], where E

_{i}_{,}

_{j}_{,}

_{k}_{≠}

*denotes the average over all*

_{j}*i*,

*j*and

*k*≠

*j*, see eq. A-5). The moments of alleles frequencies must be evaluated explicitly in order to close eq. 2. Under weak selection (i.e., retaining only first-order effects of selection), it is sufficient to evaluate such moments under neutrality only. Any effect of selection on the moments of alleles frequencies will involve at least first order effects, which will result in second or higher order effects of selection on allele frequency change. These second order effects are neglected when the change in gene frequency is evaluated to the first order only (e.g., Kirkpatrick et al., 2002; Otto and Day, 2007). We will evaluate the moments of gene frequencies in two different ways because we analyze the co-evolution of recognition markers and harming alleles under two different but complementary demographic situations:

- Infinite structured population: we assume that migration is positive (
*m*> 0) so that the evolving population is the collection of an infinite number of groups. Because the average in eq. 2 is over all groups in the population, the change of allele frequency Δ*p*_{A}at the level of the total population is deterministic. There is no genetic drift occuring at the level of the total population but genetic drift occurs locally, at the level of the deme, which generates fluctuations in gene frequencies between demes (i.e., relatedness). We then evaluate the moments of allele frequencies as is usually carried out for the infinite island model of dispersal (see the Appendix), by assuming that genetic associations (linkage-disequilibrium, relatedness) are fast variables that equilibrate before any significant change in allele frequencies (slow variables) has occurred at the level of the total population (e.g., separation of time-scale argument, Otto and Day, 2007). This approximation is valid when the strength of selection is much weaker than both the recombination and the migration rates (e.g., Nagylaki, 1993; Kirkpatrick et al., 2002; Roze and Rousset, 2008). - Finite panmictic population: we assume no migration (
*m*= 0), which is the same as considering a single panmictic population of fixed finite size*N*, where the fate of any allele in the absence of mutation is either fixation or loss from the population. Our aim is then to evaluate the probability of fixation*π*_{A}of allele A in this population. For this case we simplify the previous notation by dropping the group subscript*i*and denote by*p*_{A(}_{j}_{)}the frequency (0 or 1) of allele A in individual*j*of the population, 1 +*f*the fecundity of that individual (equivalent to eq. 1 with subscript_{j}*i*dropped) and by 1 +*f*the average fecundity in the total population (average over*N*individuals only). With these assumptions, eq. 2 boils down to$$\mathrm{\Delta}{p}_{\text{A}}={\text{E}}_{j}[({f}_{j}-f){p}_{\text{A}(j)}],$$(3)where E[·] denotes the average over all individuals in the population. Because the number of individuals is finite, Δ_{j}*p*_{A}is now a random variable. That is, eq. 3 is interpreted as the expected change in allele frequency conditional on the current realization of genotype frequencies in the population, which is a random variable because genetic drift causes the population to take different sample paths. By taking the expectation of Δ*p*_{A}over all possible sample paths one obtains the unconditional expected change in allele frequency (see eqs. A-20–A-22), which allows us to calculate directly the fixation probability*π*_{A}by evaluating exactly under neutrality the moments of allele frequencies that will appear in eq. 3 (no assumption of separation of time scale here).

### Harming versus helping

We will compare our results with those obtained previously by Rousset and Roze (2007) for the evolution of marker-based conditional helping under the same infinite-island life cycle as that described above. For conditional helping, it was assumed that individuals bearing allele H provide a benefit *B*/(*N* − 1) to a neighbour [instead of harming it with intensity *D*/(*N* − 1)], whenever actor and recipient carry the same conspicuous feature. The model for conditional helping can then be obtained directly from eq. 1 by substituting − *D* with *B*, and *p*_{R(}_{ij}_{)}(1 − *p*_{R(}_{ik}_{)})+(1 − *p*_{R(}_{ij}_{)})*p*_{R(}_{ik}_{)} with its complement *p*_{R(}_{ij}_{)}*p*_{R(}_{ik}_{)} + (1 − *p*_{R(}_{ij}_{)})(1 − *p*_{R(}_{ik}_{)}), which is the probability that individual *j* and *k* from group *i* bear identical conspicuous marker alleles. With these changes our eq. 1 is equivalent to eq. 1 of Rousset and Roze (2007).

## Results

### Infinite structured population

#### Invasion condition for harming

We find that the change in the expected frequency of the conditional harming allele H under weak selection is given by

where *H*_{R} = 2*p*_{R}(1 − *p*_{R}) is the genetic diversity in the population at the recognition locus; *F* measures the probability that two genes randomly sampled at the same locus from two different individuals from the same group are identical-by-descent (i.e., the two alleles stayed in the same deme and coalesced in that deme); (*γ*) is the probability that two pairs of genes sampled at two different loci from two (three) different individuals are identical-by-descent (see eqs. A-5–A-12). The probabilities and *γ* are measures of association between two pairs of homologous genes and are complicated functions of the demographic and genetic parameters *N*, *m*, and *r* (eqs. A-15–A-19).

The term 1 − *F* in eq. 4 is the probability that a pair of genes does not coalesce and *H*_{R}(1 − *F* ) then gives the probability that two recognition genes sampled from two different individuals are different. The term *H*_{R}(1 − *F* ) can thus be interpreted as the probability of interaction between a focal individual carrying the mutant harming allele and a recipient carrying a different marker allele from the focal individual, in which case the focal individual harms the recipient at a fecundity cost *C* to itself. The term *H*_{R}(*F* − ) can be interpreted as the probability of interaction between the focal individual and an individual carrying a harming allele identical-by-descent to that of the focal individual and a different marker allele from the focal individual (*F* − is the probability that, among two pairs of homologous genes sampled from two distinct individuals, one coalesces within the deme, but the other, here at the recognition locus, does not), in which case the focal individual is harmed and loses *D* fitness units. Finally, *H*_{R}(*F* − *γ*) can be interpreted as the probability of interaction in the focal deme between an individual, which is different from the focal individual but carries a harming allele identical-by-descent to that of the focal individual, with a third individual that carries a different marker allele from the former individual (*F* − *γ* is the probability that, among two pairs of homologous genes sampled from three distinct individuals, one pair coalesces, but the other does not), in which case the former harms the latter at a cost to the former.

With these three interaction probabilities, the term in brackets in the second line in eq. 4 can be thought of as the average frequency of interactions in the focal deme in which an actor bearing a harming allele identical-by-descent to that of the focal individual (including the focal individual himself) harms a recipient bearing a different phenotypic cue from the actor, and which causes a decrease in average patch productivity by *D* + *C*. This decrease in focal deme productivity decreases the intensity of competition faced by the focal individual’s offspring, which then have a higher chance of reaching adulthood, provided they stay in the focal deme and compete against other offspring from that deme (hence the term (1 − *m*)^{2} in eq. 4). The second line in eq. 4 is thus the fitness benefit of harming accruing to the focal individual.

If pairs of alleles sampled at the harming and recognition locus in different individuals from the same group were to coalesce independently of each other (no associations between alleles from different loci in different individuals), we would have = *γ* = *F*^{2}, in which case eq. 4 reduces to Δ*p*_{H} = − *Cp*_{H}(1 − *p*_{H})*H*_{R}(1 − *F* ) (obtained by using eq. A-15 at equilibrium). The direction of selection on the mutant allele is then given by − *C* > 0, which is the classical result for the selective pressure on an unconditional social behavior under our life-cycle assumptions (Taylor, 1992a,b). With limited dispersal (*m* < 1) and finite group size, alleles sampled at different loci from two different individuals are unlikely to be independent because the two individuals may descend from a common ancestor and may thus carry replica chromosomes. Individuals bearing recognition markers identical-by-descent are then also likely to bear harming alleles identical-by-descent; therefore > *F*^{2} and *γ*> *F*^{2}, in which case harming might eventually be selected for.

#### Harming versus helping

Using the explicit expressions for *F*, and *γ* in eq. 4 (eq. A-15–eq. A-19), we find that the mutant harming allele H might be selected for when group size is large (neglecting terms of higher order than 1/*N*) if

which is a monotonically decreasing function of *N*, *m*, and *r*. We now compare this invasion condition with that obtained for conditional helping. Substituting the explicit expressions for *F*, and *γ* into the condition for invasion of conditional helping (eq. 2 of Rousset and Roze, 2007 or eq. A-13 of the Appendix of this paper), we find that if allele H results in helping, conditional on the recipient bearing an identical recognition marker allele to the actor, it is selected for under large group size if

Since the ratio *H*_{R}/(1 − *H*_{R}) is equal to one only when the two recognition alleles are equally frequent, is smaller otherwise, and tends to zero when the frequency of one of the two recognition markers is close to fixation, the selective pressure on marker-based conditional helping is generally weaker than that on marker-based conditional harming when there are only two recognition alleles in the population. It can be shown that this result holds regardless of population size, and when population size is small the difference between the two selection pressures becomes even more pronounced. However, if there are more than two recognition alleles segregating in the population, the diversity *H*_{R} can exceed one half (with *K* neutral marker alleles, the maximal diversity is (*K* − 1)/*K*). In this case, the selective pressure on conditional helping may be stronger than that on conditional harming, but it will depend on the parameter values.

#### Invasion condition for recognition markers

Following similar calculations to those carried out in the Appendix for the harming allele (e.g., eqs. A-5–A-12), we find that the change in the frequency of the recognition allele R is given by

where *Z* is a function of *N*, *m*, *r*, which is always positive and is exactly the same as that obtained for the change in the recognition allele under conditional helping (Rousset and Roze, 2007, eq. 19 of their appendix). Hence, unless an extrinsic force maintains polymorphism at the recognition locus, selection will completely eliminate the diversity at the recognition locus, ultimately preventing the evolution of conditional harming. This is exactly the same result as that found for marker-based conditional helping and whose implications were discussed at length in Rousset and Roze (2007).

Finally, we mention that we have also investigated the joint evolution of conditional harming and recognition markers including second order effects of selection in exactly the same way as has been carried out previously for conditional helping, and which resulted in the identification of stable internal polymorphism under a restricted set of parameter values (Rousset and Roze, 2007). However, for conditional harming we have not found a stable internal polymorphism for a biologically relevant set of parameter values. This suggests that polymorphism is unlikely to be maintained by frequency-dependent selection in the case of conditional harming unless introduced by mutation.

### Finite panmictic population

#### Invasion condition for harming

We find that the fixation probability of a single mutant harming allele H, averaged over the two marker allele backgrounds in which the mutant allele might eventually appear, is given under weak selection by

where 1/*N* is the fixation probability under neutrality; *H*_{R}(0) = 2*p*_{R}(0){1 − *p*_{R}(0)} is the genetic diversity at the recognition locus when the mutant is initially introduced into the population; *P*_{1}, *P*_{2}, and *P*_{3} are functions of *N* and *r* (eqs. A-27–A-32).

The term *H*_{R}(0)*P*_{1} in eq. 8 gives the sum over the whole invasion time of the mutant allele of the probability that a focal individual carries the mutant harming allele H and interacts with another individual bearing a different marker allele from the focal individual. Therefore, *H*_{R}(0)*P*_{1} can be thought off as the counterpart of *p*_{H}(1 − *p*_{H})*H*_{R}(1 − *F* ) in the deterministic model given by eq. 4. The term *H*_{R}(0)*P*_{2} gives the sum over the whole invasion time of the probability that a focal individual bears allele H and interacts with another individual bearing allele H but a different marker allele (counterpart of *p*_{H}(1 − *p*_{H})*H*_{R}(*F* − ) in the deterministic model); that is, the alleles sampled at the harming locus in the two individuals have coalesced but those sampled at the recognition locus have not. Finally, *H*_{R}(0)*P*_{3} gives the sum over the whole invasion time of the probability that a focal individual carries allele H and a second individual carries the same allele and interacts with a third individual that carries a different marker allele from the second (counterpart of *p*_{H}(1 − *p*_{H})*H*_{R}(*F* − *γ*) in the deterministic model).

In a finite population, selection favors a mutant allele H whenever it results in a fixation probability higher than that expected under neutrality, and this might occur when the second term in eq. 8 is positive, which can be thought of as the gradient of selection on the mutant allele (force of directional selection). Hence, the direction of selection on the harming allele is independent of the initial genetic diversity *H*_{R}(0) at the marker locus. The invasion condition for the finite panmictic population model (eq. 8) is therefore qualitatively similar in form to that given by the infinite structured population model (eq. 4). Further, the interpretation of the probabilities weighting the cost *C* and damage *D* depends on qualitatively similar coalescence probabilities. In both cases an actor may interact with two categories of recipients: those that are more and those that are less likely to have inherited recognition markers and behavioral alleles from one common ancestor. This may allow an actor to discriminate between the classes of recipients if recombination is not too frequent.

In Figure 1, we graph the threshold cost-to-benefit value *C*/*D* obtained from eq. 8 below which the harming allele is selected against as a function of the recombination rate and population size (eq. A-46). This figure shows that the harming allele may be selected for under a wide range of parameter values, even if the recombination rate is large. In the absence of recombination, the threshold cost-to-benefit is approximatively given by *C*/*D* < 1/2 (eq. A-48). If recognition is perfect and the population remains of finite size, it becomes very likely that a harming allele is favored by selection.

#### Harming versus helping

If population size is large (neglecting terms of higher order than 1/*N*), conditional harming may be selected for in a finite panmictic population when

(eqs. A-46–A-47). We can again compare this invasion condition for conditional harming to that obtained for conditional helping. Results for marker-based conditional helping in a finite panmictic population with recombination have not been derived previously, and we also derived this case here (see eqs. A-49–A-50). We find that a single mutant conditional helping allele may be selected for in a finite panmictic population when population size is large if

(eq. A-51). This invasion condition depends on the initial genetic diversity *H*_{R}(0) at the recognition locus, and is qualitatively the same result as that obtained for the deterministic model for the evolution of conditional helping in an infinite structured population (Rousset and Roze, 2007).

As was also the case for the infinite structured population model, the selective pressure on conditional helping is generally weaker than that on conditional harming, but the difference is greater in the finite panmictic population case. Indeed, eq. 10 suggests that if the genetic diversity *H*_{R}(0) at the marker locus is not large enough relative to the recombination rate *r*, conditional helping can never invade, whatever the cost-to-benefit ratio. In Figure 2, we have plotted the exact threshold invasion condition as a function of the recombination rate and population size. Comparing Figure 1 and Figure 2 we see that the parameter space that favors the evolution of conditional harming is much larger than that for conditional helping. Finally, we have also evaluated the fixation probability of a mutant recognition marker, but since the resulting expression is complicated and not more informative than that for the deterministic model, we do not discuss this further.

#### Monte Carlo simulations

Because our results are accurate only to first order effects of selection (theoretically when *C*, *D*, and *B* are close to zero), we compared predicted changes in fixation probabilities due to the effect of selection with observed those using Monte Carlo simulations. Figure 3 shows a general trend for the present model: the direction of selection is very well approximated by the first-order conditions (even for *D* as large as 0.5), although the predicted fixation probabilities themselves are not accurate for strong selection.

### The relationship between conditional harming and helping

Under our life cycle assumptions, we find that marker-based conditional harming is often more likely to evolve than marker-based conditional helping (in both finite panmictic and infinite structured populations), and that the direction of selection on conditional harming is independent of the genetic diversity *H*_{R} at the recognition locus, while the direction of selection on conditional helping depends on *H*_{R} (or initial genetic diversity for the finite panmictic population model). Why is this so? In order to understand these features it is useful to express the selective pressure on conditional helping in terms of that on conditional harming.

#### Selection on helping in terms of that on harming

In order to compare the direction of selection on helping and harming, we now consider that the magnitude of the phenotypic effects on fitness under harming and helping are the same in absolute terms; namely, |*D*| = |*B*| and we use the symbol *B* to denote these effects under both cases (alternatively, we could have chosen *D*). The gradient of selection *S*_{harm} on conditional harming, which is the weight for *p*_{H}(1 − *p*_{H}) in eq. 4 and the second term in eq. 8, can then be expressed generically as

The first component, *Cκ*_{1}, can be thought of as the net inclusive fitness cost of expressing conditional harming. Under our two models, the weight *κ*_{1} (varying between zero and one) can be interpreted as the probability that the focal individual expresses harming (weighted probability over the whole invasion time for the finite panmictic population model), standardized by the average probability that an individual expresses harming in the population (including the focal individual himself) times the probability that the resulting change in patch productivity affects the competition experienced by the focal individual’s offspring. The second component, *Bκ*_{2}, in eq. 11 can be thought of as the net inclusive fitness benefit of expressing harming. In this case, the weight *κ*_{2} (varying between zero and one) can be interpreted as the average probability that an individual expresses harming in the population times the probability that the resulting change in patch productivity affects the competition experienced by the focal individual’s offspring, standardized by the probability that the focal individual gets harmed.

Because any neighbor is either identical to or different from the focal individual at the recognition locus, the probabilities that any neighbor is the recipient of actions by either a harming or a helping focal individual add up to one. It follows that the selective pressure *S* on unconditional helping (where the mutant allele codes for an act of helping of intensity *B*, which is shared equally among all neighbours, and which results in a cost *C* to the actor) can be expressed as

where *S*_{help} is the selective pressure on conditional helping and *S*_{harm}(*C*, −*B*) is the selective pressure on conditional harming (eq. 11), where the argument *B* is replaced by −*B* in order to account for the fact that all categories of recipients are helped. The equation for *S* simply splits the interaction frequencies of a focal individual helping patch members unconditionally into two classes: those that involve individuals carrying identical marker alleles to it and those that involve individuals carrying different marker alleles.

Suppose that the selective pressure on unconditional helping is equal to zero (*S* = 0), but that individuals would benefit from helping recipients carrying identical marker alleles (*S*_{help} > 0). Then, the net inclusive fitness benefit for providing help to on category of individuals is exactly compensated by an inclusive fitness loss for providing help to the other category. Hence, harming the latter category of individuals would actually increase inclusive fitness. In this special case, the inclusive fitness benefit obtained by helping one category of individuals is exactly equal to the inclusive fitness benefit that would be obtained by harming the other category. This does not imply that the interaction frequencies with the two categories of individuals are equal. On the contrary, when there are only two recognition marker alleles, we have *H*_{R} ≤ 1/2, so that individuals are much more likely to interact with others bearing identical marker alleles than with those bearing different marker alleles. By contrast, when there more than two recognition alleles, say *K*, then *H*_{R} ≤ (*K* − 1)/*K*, in which case it becomes possible that individuals interact more often with others bearing different marker alleles.

In general, the selective pressure on unconditional helping is unlikely to be equal to zero (*S* ≠ 0), in which case it can be expressed as

which, by comparing with eq. 11, shows that whether the inclusive fitness benefit of conditional helping will be larger than that under conditional harming depends on whether the inclusive fitness benefit of unconditional helping is greater than zero. From eq. 11 and eq. 13, the selective pressure on conditional helping is stronger than that on conditional harming when

#### Comparing the intensities of selection

For the infinite structured population model, we have *S* = − (1 − *F*)*C* (eq. A-14 of the Appendix), so that the inclusive fitness benefit under both unconditional helping and harming are exactly the same. From eq. 4, one can see that 0 ≤ *κ*_{1} ≤ (1 − *F* )*H*_{R}; the inclusive fitness cost will never exceed (1 − *F*)*H*_{R} for this model. When there are only two recognitions markers (*H*_{R} ≤ 1/2) the inclusive fitness cost under conditional helping is equal to or greater than that under harming. Then, eq. 14 can never be satisfied and the total selective pressure on conditional harming will either be equal to or stronger than that on conditional helping (see also eq. 5 and 6). But the situation can be reversed when there are more than two recognition alleles. In this case, the inclusive fitness cost under conditional helping may be weaker than that under harming because individuals may interact more frequently with recipients carrying different marker alleles (*H*_{R} ≤ (*K* − 1)/*K* with *K* marker alleles), and eq. 14 can now be satisfied (see also eq. 5 and 6).

For the finite panmictic population model, we have *S* = − *C* − (*B* − *C*)/*N* (see eq. A-49), so that the inclusive fitness benefit under both unconditional helping and harming no longer cancel each other out. This stems from the fact that when interactions are unconditional, harming is selected for in a finite panmictic population (Hamilton, 1971), while the sign of the behavioral effect on neighbors does not affect the direction of selection in the infinite structured population model. Everything else being equal, the dice are thus more loaded in favor of conditional harming than in favor of conditional helping in finite panmictic populations (e.g., compare eq. 5 and eq. 9)

More generally, the above arguments allowing us to contrast the selective pressure on helping and harming should apply more widely. They should apply whenever a focal individual may express conditionally helping or harming with complementary probabilities over all the events it might face throughout the life-cycle and when the selective pressure on conditional harming can be expressed as eq. 11. For instance, this is the case when a focal individual helps patch members only when it is a native and harms patch members only when it is an immigrant. It was then found in models with similar baseline life-cycle as ours (namely where *S* = − (1 − *F*)*C*) that the selective pressure on conditional harming is often, but not always, stronger than that on conditional helping (Lehmann, 2003; El Mouden and Gardner, 2008). In this case, one has *κ*_{1} = *m*{1 − (1 − *m*)^{2}/*N* } and *κ*_{2} = *m*(1 − *m*)^{2}/*N* because an individual expresses harming only when it disperses, in which case no other individuals carries a harming allele identical-by-descent to that of the focal individual, which is then not harmed more often than average (substituting these two expressions into eq. 11 and eq. 13 allows one to recover, respectively, eq. 19 and eq. 12 of El Mouden and Gardner, 2008). These expressions allows us to show that eq. 14 can be satisfied only if migration is strong because in that case kin competition becomes negligible, while there may still be benefits from helping relatives as long as relatedness remains positive.

In the same vein as just discussed, one may assume that a focal individual perfectly recognizes and helps only individuals born in its natal deme and harms all others, in which case it can again be shown that for a large set of parameter values the selective pressure for conditional harming is stronger than that for conditional helping (Lehmann, 2003; *κ*_{1} = {(1 − *m*)*m* + *m*} − (1 − *m*)^{2} [{(1 − *m*)*m* + *m*}/*N* + *m*(1 − *m*)^{2}*F*^{R}(*N* − 1)/*N*] and *κ*_{2} = (1 − *m*)^{2} [{(1 − *m*)*m* + *m*}/*N* + *m*(1 − *m*)^{2}*F*^{R}(*N* − 1)/*N*}, where (1 − *m*)*m* + *m* is the probability that a focal individual interacts with another individual that is not born in its natal deme, *m*(1 − *m*)^{2} is the probability that the focal individual is a native and that another native individual interacts with a third individuals that is a non-native, and *F*^{R} = 1/*N* + (*N* − 1)*F*/*N* is the probability of identity between homologous genes sampled in two native individuals). Finally, we mention that Johnstone and Cant (2008) analyzed a patch-structured model with overlapping generations and sex-biased dispersal of juveniles, where selection on harming was also found to be stronger than that on helping for a large set of parameter values, but in their case the behavior was expressed unconditionaly so that the above considerations do not apply to their model.

#### The effect of *H*_{R} on helping and harming

Equations 4 and 8 show that the force of selection *S*_{harm} on marker-based conditional harming is proportional to the amount of genetic diversity *H*_{R} in the population at the recognition locus (or initial genetic diversity for the panmictic population model). This stems from the fact that an interaction between an actor-recipient pair is conditional on the probability that the two individuals carry different marker alleles. This probability is proportional to the genetic diversity *H*_{R} because genetic drift reduces diversity within demes (or total population diversity for the finite panmictic population model) by a fractional amount in each generation, until equilibrium is eventually reached (Crow and Kimura, 1970; Hartl and Clark, 2007). Consequently, the interaction probabilities between actor and recipient determining *κ*_{1} and *κ*_{2} are proportional to *H*_{R} (e.g., eq. 4). It follows that the magnitude of population diversity (or initial diversity for the panmictic population model) should not affect the direction of selection on conditional harming, and eq. 4 also holds in the case where there is an arbitrary number of recognition alleles segregating in the population (in which case
${H}_{\text{R}}=1-{\sum}_{h}{p}_{\text{R},h}^{2}$, where *p*_{R,}* _{h}* is the frequency of the

*h*th neutral marker allele, see eq. A-5; notice that we did not demonstrate that this result also holds for the panmictic population case).

By contrast to harming, the direction of selection on marker-based conditional helping was found to depend on the magnitude of genetic diversity *H*_{R} at the recognition locus (Rousset and Roze, 2007, eq. 2 or eq. A-13 in the Appendix, and see eq. A-49 for the panmictic population model). In this case, the interaction between an actor and a recipient within a deme is conditional on the probability that the pair carry identical marker alleles, which is the complement of the probability that they carry different alleles. This probability is typically larger but not proportional to the total population probability of identity, given by 1 − *H*_{R} (see eq. A-13), because in each generation genetic drift adds new genetic identity to existing one through the action of coalescence (Crow and Kimura, 1970; Hartl and Clark, 2007). Thus, the direction of selection on conditional helping will not be proportional to 1 − *H*_{R} but will rather be affine in *H*_{R} (say *α* + *βH*_{R}). This can be seen from eq. 11 and eq. 13: if the direction of selection on conditional harming, *S*_{harm}, is proportional to *H*_{R}, the direction of selection on *S*_{help} will depend on the value of *H*_{R}.

## Discussion

Natural selection favors those genotypes that confer on their carriers the highest lifetime reproductive success (fitness defined here as the expected number of offspring that reach the stage of reproduction) because these genotypes are more likely to introduce replicate copies of themselves into the next generation than alternative genotypes. There are two basic and very different means by which a mutant allele can cause its carriers to have a higher fitness than those individuals bearing an alternative, resident allele. Either the mutant confers higher vital rates on its carriers (higher fecundity or survival) or the mutant confers lower vital rates to non-carriers. The latter case can be defined as harming, and when it occurs it may decrease the intensity of competition experienced by a carrier of the harming allele, or that experienced by its offspring.

### Evolution of marker-based conditional harming

We have analyzed the joint evolution of neutral recognition markers and marker-based conditional harming under two different but complementary demographic scenarios: finite panmictic and infinite structured populations. Our results show that for a mutant harming allele to be selected for under these two scenarios, the costs to an actor of expressing harming and being harmed must be offset by the benefits obtained from the reduction in competition faced by the actor’s offspring, which is due to all actors in the population expressing conditional harming (eq. 4 and eq. 8). The first cost (cost of harming) depends on the probability that an actor interacts with another individual bearing a different marker allele from itself; the second cost (cost of being harmed) depends on the probability that an actor interacts with another individual bearing the harming allele and a different marker allele from the actor; finally, the benefit of harming depends on the probability that in the population of a focal actor, actors (including the focal actor) interact with other individuals bearing different marker alleles from those of the actor.

The interaction probabilities that weight the costs and benefits of harming depend on population size, *N*, recombination rate, *r*, and migration rate *m* (for the structured population case). Total finite population size (or finite local group size with limited dispersal) results in genetic drift, which entails that an actor may interact with other individuals that have the same common ancestor as the actor (i.e., coalescence of alleles sampled in different individuals occurs). These individuals are then likely to carry the mutant harming allele and the same marker allele as the actor. Interaction with such individuals will reduce the costs of being harmed, but also the benefits, because fewer individuals are likely to be harmed in the population (or local group). In the absence of recombination, the benefits might exceed the costs because an individual bearing the harming allele will never be harmed (perfect recognition), but as long as there is genetic variation in the population, individuals bearing the resident allele are harmed, which results in a decrease in competition felt by the actor or its offspring. Recombination increases the cost of being harmed, because the descendants of an ancestor bearing the harming allele may carry different marker alleles, but it also increases the benefits of harming for the same reason, as more individuals are harmed in the population.

Our results show that the selective pressure on marker-based conditional harming is a decreasing function of the three parameters *N*, *m*, and *r* (eq. 6). This is qualitatively exactly what is usually found for the selective pressure on unconditional helping in a spatially subdivided population with the island model of dispersal (e.g., Eshel, 1972; Aoki, 1982; Rogers, 1990; Taylor and Irwin, 2000; Gardner and West, 2006), and is also what was found for the invasion of marker-based conditional helping (Rousset and Roze, 2007). But our results also suggest that the selective pressure on conditional harming can be stronger than that on conditional helping under otherwise similar life-cycle assumptions. This result holds for both our finite panmictic and infinite structured population scenarios (compare eq. 5 and eq. 6, and eq. 9 and eq. 10). The difference between the selective pressure on conditional harming and helping is actually expected to be greatest in finite panmictic populations (see section “The relationship between conditional harming and helping”). We then observe that a single mutant harming allele can be selected for under a wide range of parameter values, even in the presence of recombination (Fig. 1).

Under our life cycle assumptions, the selective pressure on conditional harming is often stronger than that on conditional helping because the inclusive fitness benefit obtained through conditional harming and helping (total effect through *B* and *D*, assuming they are of similar magnitude) are identical in the infinite structured population case, and larger under harming for the panmictic population case (see section “The relationship between conditional harming and helping”). But at the same time, the inclusive fitness cost of expressing harming (total effect through *C*, eq. 11) can be lower than that of expressing helping. In particular, when there are only two recognition markers segregating in the population, individuals tend to interact more often with others having identical marker alleles (*H*_{R} ≤ 1/2) and thus pay the direct cost of expressing conditionally helping more often than they would if they expressed harming conditionally. When there are more marker alleles segregating at the recognition locus (*H*_{R} ≤ (*K* − 1)/*K* with *K* marker alleles), the selective pressure on conditional helping can become stronger than that on conditional harming as individuals tend to pay the direct cost of harming more often than that of helping (e.g., eq. 5 and 6). Different life-cycle assumptions might also lead to different selective regimes on conditional harming. For instance, in patch-structured populations with maternally transmitted symbionts spreading through host populations by hampering the reproduction of uninfected females (cytoplasmic incompatibility), the condition for invasion of harming was found to be a non-monotonic, dome shaped function of group size in a model with similar basic structure to ours (Reuter et al., 2008). It is also well-known that introducing overlapping generations with only juvenile dispersal into the type of models considered here can increase the selective pressure on unconditional helping (*S* > 0 in eq. 13, Taylor and Irwin, 2000; Irwin and Taylor, 2001), which may tip the balance in favor of conditional helping instead of conditional harming, although this is likely to depend on the life-cycle parameter values (Johnstone and Cant, 2008).

### Harming versus spite

The selective pressure on a mutant harming allele depends on how its expression benefits its carrier and the carrier’s relatives by reducing the fecundity of individuals bearing the alternative allele. Hamilton (1970) called *spiteful* a behavior decreasing the fitness of the actor and that of the recipient of the act of harming. We mention that harming might qualify as spiteful *sensu* Hamilton (1970) in both our finite panmictic and infinite structured population models. This can be seen by noting that the net change in the fitness of a carrier due to it expressing the harming allele (and holding everything else constant) can be obtained for the infinite structured population model by setting *H*_{R}(*F* − ) and *H*_{R}(*F* − *γ*) equal to zero in eq. 4 and for the finite panmictic population model by setting *P*_{2} and *P*_{3} equal to zero eq. 8. In both cases, there is a wide range of a parameter values (*D*, *C*, *m*, *N*, and *r*) where the resulting change in fitness can be negative (hence expressing the mutant allele results in a net fitness cost to the carrier when everything else is hold constant) but the harming allele will still be favored by selection because these direct costs are offset by the reduction in competition felt by the offspring of relatives of the focal individual. However, the conditions under which harming qualifies as spiteful in the sense of Hamilton (1970) are complicated and do not help us here to further understand the ecological and demographic conditions under which conditional harming is selected for, so that we did not present such computations.

### Marker-based recognition, green-beards, and mate choice

Marker-based recognition in panmictic populations is sometimes referred to as the green-beard mechanism: a conspicuous phenotypic effect of a gene is recognized by other individuals bearing that gene, and where that gene also has a pleiotropic effect on the behavior of individuals expressing the conspicuous phenotype (Dawkins, 1982). Strictly speaking our marker-based conditional harming and helping models for both finite panmictic and infinite structured populations correspond to kin-recognition mechanisms. This is so because in both cases the marker alleles are exchangeable (whether the harming or helping alleles arise on an R or r marker allele background does not affect discrimination). By contrast, under the green-beard mechanism (Dawkins, 1982), a particular marker allele is postulated to be associated with a specific behavioral phenotype so that conspicuous markers are not exchangeable (see below for an empirical example).

Independently of the exact nature of the marker-based recognition mechanism (e.g. kin recognition or green beard), our results suggest that the evolution of marker-based helping in panmictic populations may be selected for only under small population size, while marker-based harming might evolve under a much larger set of parameter values. In the light of this observation it is interesting that the compelling documented examples of green-beards in natural populations are of the harming type. Indeed, the green-beard found in the red fire ant *Solenopsis invicta* is of this type (Keller and Ross, 1998), where workers homozygous for allele b at the Gp-9 locus kill those individuals that do not contain it (BB queens) while not inducing killing of individuals that do (Bb queens). Note that here allele B cannot be exchanged with allele b without affecting discrimination. Other examples of marker-based conditional harming may be found among bacterial strains. Some bacteria release into their environment intra-specific antagonistic compounds such as bacteriocins and bacteriophages, which allows them to suppress the growth of competing strains (Riley and Gordon, 1999; Gardner et al., 2004). Recognition in this case is molecular with the bacteriocin gene tightly linked to specific immunity genes that block the effect of the bacteriocins; and molecular discrimination may even occur between carriers and non-carriers of isogenic phages (Brown et al., 2006). For bacteria with recurrent cycles of colonization and population growth, the number of founding clones will probably be more relevant than the stationary population size to describe the change in genotype frequency in the population; in that case our parameter *N* can be thought of as the number of founding clones.

Another situation where marker-based recognition can be used to discriminate between categories of recipients is in the context of assortative mating (Crow and Kimura, 1970; Kirkpatrick, 1982; Seger, 1985). Females (or males) could prefer to mate with those individuals of the opposite sex that carry identical marker alleles to them at an arbitrary recognition locus that has no direct effects on fitness (Castro and Toro, 2006). There are several similarities between our models and the mate choice model of Castro and Toro (2006). These authors also consider that individuals carry two loci: one where arbitrary recognition alleles segregate and another that codes for mating expressed conditionally on pairs of individuals bearing identical marker alleles at the recognition locus; the result is that individuals are more likely to interact with relatives. Castro and Toro then show by simulations that the spread of a choice allele resulting in females mating only with males carrying identical markers is enhanced by finite population size effects (whether the population is panmictic or structured), which corroborates our own results. However, our formalization does not apply directly as it stands to mate choice. By contrast to the model of Castro and Toro, we do not consider a process by which individuals search for others carrying identical (or different) recognition markers. Such a search process could be included in our models by introducing different acceptance probabilities for individuals bearing identical or different marker alleles, so that individuals would stop searching once they have found a partner they accept to interact with (or mate with in the context of mate choice). This deserves further formalization, especially since mate choice is also likely to depend on inbreeding, an inevitable consequence of finite patch or population size.

### Conclusion

Harming behaviors may not be uncommon in nature. For instance, segregation distorter alleles may produce toxins during meiosis to which they but not their alternatives are resistant; the distorter thus increases in frequency by reducing competition for fertilization (e.g., Lyttle, 1991; Ridley, 2003; Burt and Trivers, 2006). Maternally transmitted symbionts can spread through host populations by hampering the reproduction of uninfected females, thereby reducing competition for symbiont carriers (e.g., Werren, 1997; Ridley, 2003; Burt and Trivers, 2006). In all these cases a mutant allele spreads by harming others and this functions because the interaction neighborhood is small enough that the reduction of vital rates of others due to the behavior of the actor, or that of its relatives, decreases the intensity of competition experienced by the actor or its offspring (the interaction neighborhood is actually very small for segregation distorters).

In addition to the results reported here, several models have already identified ecological and demographic conditions for the evolution of harming behaviors in structured populations, where localized migration generates small interaction neighborhoods (Gardner et al., 2004; Lehmann et al., 2006; Gardner et al., 2007; Lehmann et al., 2007a; Johnstone and Cant, 2008; El Mouden and Gardner, 2008). All these results broaden the scope of biological situations where harming may occur. They show not only that harming might evolve in both finite panmictic and structured populations, but suggest that, under certain situations, harming is actually more likely to evolve than helping. This should encourage behavioral ecologists to seek evidence for conditional harming rather than conditional helping.

## Acknowledgments

We thank Andy Gardner for making helpful comments on the manuscript, and are grateful to Nicholas Barton for unleashing the brakes of a congested review and for its helpful suggestions on the manuscript. This work was supported by a grant from the Swiss NSF to LL and by NIH grant GM28016 to MWF. This is publication ISEM 0X-XXX.

## Appendix

#### Infinite structured population

##### Gene frequency change

The expected change in the average frequency *p*_{A} of allele A over one generation can be written under our life cycle as

where *w _{ij}* is the expected number of offspring of individual

*j*from group

*i*that will reach the next adult generation. This is the Price equation (Price, 1970; Hamilton, 1970), but since the population is assumed to be of constant size, the mean fitness is equal to one (E

_{i}_{,}

*[*

_{j}*w*] = 1). With our notations introduced in the main text, the fitness of individual

_{ij}*j*from group

*i*can be written as

(Roze and Rousset, 2005; Lehmann et al., 2007b; Rousset and Roze, 2007). Assuming that *C* and *D* are of small order *δ*, a Taylor expansion of *w _{ij}* substituted into eq. A-1 produces eq. 2 of the main text.

##### Mutant harming allele

We derive here the change in the expected allele frequency Δ*p*_{H} of the harming allele when evolution occurs in the infinite island model of dispersal. The derivation of the change in the expected frequency of a recognition allele is carried out in exactly the same way.

Our analysis closely follows that of Rousset and Roze (2007) for marker-based conditional helping. We also assume that *K* alleles, denoted *R*_{1}, *R*_{2}, … *R _{K}*, may segregate at the recognition locus (

*K*= 2 in the main text). We denote by

**x**

_{R(}

_{ij}_{)}(

*p*

_{R,1(}

_{ij}_{)},

*p*

_{R,2(}

_{ij}_{)}, …,

*p*

_{R,}

_{K}_{(}

_{ij}_{)}) the vector with

*p*

_{R,}

_{l}_{(}

_{ij}_{)}being the frequency (0 or 1) of recognition marker

*l*in individual

*j*from group

*i*. Hence, the element

*l*of this vector is equal to one if individual

*j*from group

*i*bears allele

*R*, zero otherwise. With this, the effect of social interactions on the relative fecundity of individual

_{l}*j*from group

*i*(i.e., relative fecundity without the baseline fecundity unit) can be written as

where · denotes the dot product, and with only two alleles segregating in the population this equation reduces to eq. 1 of the main text. The average of this equation over all individuals in group *i* is then given by

whose average over all *i* gives *f* (*f* = E* _{i}*[

*f*]).

_{i}Inserting eq. A-3 and eq. A-4 into eq. 2 allows us to write

where *p*_{H(}_{i}_{)} is the average of *p*_{H(}_{ij}_{)} over all individuals within group *i*, and

where E_{i}_{,}_{j}_{,}_{k}_{≠}* _{j}*[·] denotes the average over all

*i*,

*j*and

*k*≠

*j*. By substituting

*B*for −

*D*and

**x**

_{R(}

_{ij}_{)}·

**x**

_{R(}

_{ik}_{)}for (1 −

**x**

_{R(}

_{ij}_{)}·

**x**

_{R(}

_{ik}_{)})in eq. A-5 and eq. A-6 one obtains the equation for conditional helping (Rousset and Roze, 2007, eqs. 6–7 of their appendix).

In order to close the equation for Δ*p*_{H}, we need to evaluate under neutrality all the moments of the form E_{i}_{,}_{j}_{,}_{k}_{≠}* _{j}*[·] appearing in eq. A-5. This derivation has already been detailed in the earlier work on conditional helping (Rousset and Roze, 2007, p. 3–6 of their appendix). For this reason, we present only the results we need here, without providing the derivations. In order to evaluate eq. A-5, we first need

where *F* is the probability that two genes randomly sampled at the same locus from two different individuals are identical-by-descent (e.g., they stayed in the same deme and coalesced in that deme). We also need

and

where ${H}_{\text{R}}=1-{\sum}_{h=1}^{K}{p}_{\text{R},h}^{2}$ is total genetic diversity in the population. We further need

where is the probability that two pairs of genes sampled at two different loci from two different individuals are identical-by-descent. Finally, we need

where *γ* is the probability that two pairs of genes sampled at two different loci from three different individuals are identical-by-descent.

On substitution of eqs. A-7–A-11 into eq. A-5 we find after simplification that the change in frequency of the mutant harming allele can be written as

which is eq. 4 of the main text.

##### Mutant helping allele

For comparative analysis, we recall the results obtained for the change in frequency of the mutant allele H when it results in conditional helping (Rousset and Roze, 2007, p. 11 of the appendix). Substituting *B* for −*D* and **x**_{R(}_{ij}_{)} · **x**_{R(}_{ik}_{)} for (1 − **x**_{R(}_{ij}_{)} · **x**_{R(}_{ik}_{)}) in eq. A-5 and eq. A-6, and using again eqs. A-7–A-11, the change in the frequency of allele H for conditional helping can be written as

which is eq. 1 of Rousset and Roze (2007) written a bit differently. The term 1 − *H*_{R}(1 − *F*) in this equation can be interpreted as the probability of interactions between a focal individual carrying the mutant helping allele and a recipient carrying an identical marker allele to that of the focal individual. The term *F* − *H*_{R}(*F* − ) can be interpreted as the probability of interaction between the focal individual and an individual carrying a helping allele identical-by-descent to that of the focal individual and an identical marker allele. Finally, *F* − *H*_{R}(*F* − *γ*) can be interpreted as the probability of interaction in the focal deme between an actor, which is different from the focal individual but carries a helping allele identical-by-descent to that of the focal individual, with a third individual who carries an identical marker allele to that of the actor.

When the diversity at the recognition locus in a deme is zero (*H*_{R} = 0), eq. A-13 reduces to

where the term in parentheses is the classical selective pressure on unconditional helping derived by Taylor (1992a), and after simplification (using eq. A-15 at equilibrium) further reduces to Δ*p*_{H} = −*p*_{H}(1 − *p*_{H})*C*(1 − *F* ). Comparing eqs. A-12–A-14 illustrates that the gradient of selection on conditional helping is equal to the gradient of selection on unconditional helping minus the gradient of selection on conditional harming when *D* is replaced with −*B* in eq. A-12. This result holds regardless of the specificities of the life cycle and follows from eq. 1 (or eq. A-5) because the frequency of interaction, **x**_{R(}_{ij}_{)} · **x**_{R(}_{ik}_{)}, with individuals bearing similar recognition alleles (conditional helping) is exactly one minus the frequency of interaction, 1 − **x**_{R(}_{ij}_{)} · **x**_{R(}_{ik}_{)}, with individuals bearing dissimilar recognition alleles (conditional harming), so that the sum of interactions over the two cases is equal to that occurring when the trait is expressed unconditionally.

##### Probabilities of identity-by-descent

In order to evaluate the change of allele frequency Δ*p*_{H} explicitly (either the harming our the helping allele), it remains to evaluate *F*, and *γ*, which can be obtained by writing down recursion equations for these variables (e.g., Kimura, 1963; Karlin, 1968, Rousset and Roze, 2007, p. 10 of their appendix). The recursion for *F* is given by

The recursions for and *γ* depend on *δ*, which is the probability that two pairs of genes sampled at two different loci from four different individuals are identical-by-descent, and they satisfy

where

Solving these recursions gives complicated expressions for , *γ* and *δ*, in terms of *N*, *m* and *r*. Substitution of the resulting expressions into eq. A-12 and eq. A-13, and assuming large population size produces eq. 5 and eq. 6 of the main text.

#### Finite panmictic population

##### Probability of fixation

In this appendix we give a very brief summary of the argument developed in earlier work to compute the probability of fixation of a mutant allele under weak selection in a one-locus setting (Rousset, 2003; Lessard and Ladret, 2007), and which can directly be applied to a multilocus setting (Lehmann and Rousset, 2009).

From eq. 3 the change of allele frequency at time *t* is given under weak selection by

where we have now made explicit that the change in allele frequency is conditional on the vector **p**(*t*) of genotypes frequencies in the population at time *t*. Call Pr(**p**(*t*)) the distribution of **p**(*t*) at time *t*, conditional on the initial state **p**(0) of the population. We will use the expectation operator E [·] without subscripts to denote an expectation over the distribution Pr(**p**(*t*)) in the population, e.g., E [*p*_{A}(*t*)] = Σ_{p}_{(}_{t}_{)} Pr(**p**(*t*)) *p*_{A}(*t*). With this and eq. A-20, the expected unconditional change of allele frequency at time *t* can be written as

from which the fixation probability *π*_{A} of allele A can be obtained as

where *p*_{A}(0) is the initial frequency of the mutant in the population (Rousset, 2003; Lessard and Ladret, 2007; Lehmann and Rousset, 2009).

The problem is thus to obtain a closed form solution to eq. A-21. This can be obtained by noting that the term E* _{j}*[(

*f*−

_{j}*f*)

*p*

_{A(}

_{j}_{)}(

*t*)|

**p**(

*t*)] can be expressed, as in the previous infinite structured population model, as a sum of selection coefficients (e.g.

*C*,

*D*, (1 −

*m*)

^{2}

*C*, etc.), each weighted by averages of products of allele frequencies sampled from the same or different loci, from the same or different individuals (e.g. E

_{j}_{,}

_{k}_{≠}

*[*

_{j}*p*

_{H(}

_{j}_{)}[

*p*

_{H(}

_{k}_{)}], E

_{j}_{,}

_{k}_{≠}

*[*

_{j}*p*

_{H(}

_{j}_{)}

*p*

_{R(}

_{k}_{)}], see eq. A-5). These moments must then be integrated over the distribution Pr(

**p**(

*t*)). But since any effect of selection on the distribution Pr(

**p**(

*t*)) will be at least of first order, any effect of selection on this distribution will result in second or higher order effects of selection on allele frequency change. Hence, in order to evaluate the first order effect of selection on allele frequency change, it is sufficient to consider this distribution under neutrality only. Practically, this consists of evaluating expectations of products of allele frequencies under neutrality in the same way as was carried out in the last section, and more generally in population genetics (e.g., Kimura, 1963; Karlin, 1968; Crow and Kimura, 1970).

In order to evaluate these moments we use the notation for multilocus models in infinite populations from earlier work (Kirkpatrick et al., 2002; Roze and Rousset, 2008) with some specificities to account for total finite population size (Lehmann and Rousset, 2009). We will use the expectation operator notation without brackets (e.g.,
${\text{E}p}_{\text{A}}^{\circ}(t)\equiv {\text{E}}^{\circ}[{p}_{\text{A}}(t)]$), where the superscript signifies that the expectation is evaluated in the neutral process. Because alleles can be sampled from different individuals, we denote the expectation of an average of products of sets of allele frequencies sampled in different individuals by
${\text{E}p}_{S}^{\circ}$ with *S* *S*_{1}/*S*_{2}/…/*S*_{|}_{S}_{|}, where each *S _{j}* is a set of alleles sampled from the same individual, the “/” symbol separates sets of alleles sampled from distinct individuals, and |

*S*| is the total number of different individuals from which sets of loci have been sampled. For instance,

is the probability that a randomly sampled individual from the population carries allele H at the helping locus and allele R at the recognition locus (expectation of the frequency of gamete HR); and

is the probability that, among two distinct randomly sampled individuals in the population, one has chromosome HR, while the other individual carries allele R.

##### Mutant harming allele

In this appendix, we derive the average fixation probability of the harming allele H. Because the analysis for finite populations turns out to be more complicated than for infinite populations we assume, for simplicity, that only two alleles segregate at the recognition locus. Dropping subscript *i* in eq. 1 gives

and

On substitution of these equations into eq. A-20 we have

By taking the expectation of this unconditional change in allele frequency and using the notations introduced above, one obtains the conditional change of allele as

where

and

Inserting the last two equations into eq. A-28 produces

On substitution of this equation into eq. A-22, one can then evaluate the first-order Taylor polynomial for the fixation probability of a mutant harming allele conditional on some initial genotype distribution given by **p**(0) = (*p*_{HR}(0), *p*_{Hr}(0), *p*_{hR}(0), *p*_{hr}(0)), where *p _{V}* (0) is the initial frequency of gamete

*V*.

A single mutant harming allele (*p*_{H}(0) = 1/*N* ) can initially arise on either of the two marker-allele backgrounds at the recognition locus, and our aim is to evaluate the average _{A} fixation probability of a single mutant harming allele H, averaged over the two marker allele backgrounds in which the mutant allele could appear. A single initial copy of the harming allele appears on an R background with probability *p*_{R}(0) in which case the initial gamete frequencies in the population are given by **p**_{R}(0) = (*p*_{HR}(0) = 1/*N*, *p*_{Hr}(0) = 0, *p*_{hR}(0) = *p*_{R}(0) − 1/*N*, *p*_{hr}(0) = 1 − *p*_{R}(0)). The same copy of the mutant appears on the alternative background with probability 1 − *p*_{R}(0) in which case the initial gamete frequencies in the population is given by **p**_{r}(0) = (*p*_{HR}(0) = 0, *p*_{Hr}(0) = 1/*N*, *p*_{hR}(0) = *p*_{R}(0), *p*_{hr}(0) = 1 − *p*_{R}(0) − 1/*N* ). With this, the first order Taylor polynomial of the average _{A} fixation probability of a single mutant harming allele can be written as

where

##### Probabilities of identity-by-descent

In order to evaluate the average fixation probability _{A} explicitly, it now remains to evaluate the neutral moments
${\text{E}p}_{\text{HR}}^{\circ},{\text{E}p}_{\text{H}/\text{R}}^{\circ},{\text{E}p}_{\text{HR}/\text{H}}^{\circ},{\text{E}p}_{\text{HR}/\text{R}}^{\circ},{\text{E}p}_{\text{H}/\text{H}/\text{R}}^{\circ},{\text{E}p}_{\text{HR}/\text{HR}}^{\circ},{\text{E}p}_{\text{HR}/\text{H}/\text{R}}^{\circ}$, and
${\text{E}p}_{\text{H}/\text{H}/\text{R}/\text{R}}^{\circ}$. These moments will be affected by reproduction and recombination, and we evaluate them again by using standard methods (e.g., Kimura, 1963; Karlin, 1968; Crow and Kimura, 1970). Only moments involving alleles sampled from the same individual at the two different loci will be affected by recombination because these alleles may descend from different individuals before recombination, and only moments involving genes sampled from different individuals may be affected by reproduction because coalescence of these genes may occur. Over the recombination phase
${\text{E}p}_{\text{H}/\text{R}}^{\circ},\text{E}{p}_{\text{H}/\text{H}/\text{R}}^{\circ}$, and
${\text{E}p}_{\text{H}/\text{H}/\text{R}/\text{R}}^{\circ}$ remain constant, and the remaining expectations change according to the recursions

while over the reproduction phase, one has

By solving the above equations with the initial genotype distribution given by **p**(0), we can then evaluate *P*_{1}, *P*_{2}, *P*_{3}. We find that *P*_{1} = *N*/(*N* − 1), whereas *P*_{2}, and *P*_{3} are complicated functions of *N* and *r* but they are independent of the frequency of the marker alleles; that is *H*_{R}(0) factors out of the numerator in eqs. A-33–A-35 and is thus cancelled by the denominator. To the leading order in 1/*N* we have *P*_{1} = 1,

and

##### Invasion condition for the mutant harming allele

The expressions for *P*_{1}, *P*_{2}, *P*_{3} allow us to evaluate eq. A-32 explicitly. After simplification carried out with Mathematica (Wolfram, 2003), we find that the average fixation probability of the mutant harming allele can be expressed as

where

and

From eq. A-42, the threshold cost-to-benefit ratio under which a mutant harming allele is selected against is then given by

When group size *N* is large, eq. A-46 simplifies to

which gives eq. 9 of the main text. In the absence of recombination (*r* = 0), the threshold cost-to-benefit ratio under which the harming allele is selected against is given by

##### Invasion condition for the mutant helping allele

Substituting −*D* with *B* and (*p*_{R(}_{ij}_{)} − *p*_{R(}_{ik}_{)})^{2} with 1 − (*p*_{R(}_{ij}_{)} − *p*_{R(}_{ik}_{)})^{2} in eq. A-27, and using an analogous argument to that above, one can evaluate the average fixation probability of a single mutant allele expressing helping conditionally on both individuals bearing identical phenotypic markers. But the change of fixation probability can also be obtained by using eq. A-42 and eq. 12 of the main text. Replacing *D* with *B* in eq. A-42, one has *S*_{harm} = *H*_{R}(0) (*BX*_{1} − *CX*_{2})/*X*_{3}. For this model, one also has *S* = − *C* − (*B* − *C*)/*N*, which is the perturbation of the fixation probability of a mutant allele expressing unconditional helping towards neighbors in a finite panmictic population (Rousset, 2004; Lehmann et al., 2007a), and was anticipated by Hamilton (1971). With this and using *S*_{help} = *S* − *S*_{harm}(*C*, −*B*), the average fixation probability of a single mutant allele expressing helping conditionally can then be expressed as

The average fixation probability given by eq. A-49 then allows us to evaluate the threshold cost-to-benefit ratio below which the helping allele is selected against as

which more complicated that the invasion condition on conditional harming because it involves the function *X*_{3}. When group size *N* is large, the threshold reduces to

which is eq. 10 of the main text. In the absence of recombination (*r* = 0), the threshold cost-to-benefit ratio below which the helping allele is selected against is given by

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- Evolution of helping and harming in heterogeneous groups.[Evolution. 2013]
*Rodrigues AM, Gardner A.**Evolution. 2013 Aug; 67(8):2284-98. Epub 2013 Apr 13.* - Evolution of helping and harming in heterogeneous populations.[Evolution. 2012]
*Rodrigues AM, Gardner A.**Evolution. 2012 Jul; 66(7):2065-79. Epub 2012 Mar 4.* - Sex differences in dispersal and the evolution of helping and harming.[Am Nat. 2008]
*Johnstone RA, Cant MA.**Am Nat. 2008 Sep; 172(3):318-30.* - How life history and demography promote or inhibit the evolution of helping behaviours.[Philos Trans R Soc Lond B Biol Sci. 2010]
*Lehmann L, Rousset F.**Philos Trans R Soc Lond B Biol Sci. 2010 Sep 12; 365(1553):2599-617.* - The evolution of growth trajectories: what limits growth rate?[Biol Rev Camb Philos Soc. 2011]
*Dmitriew CM.**Biol Rev Camb Philos Soc. 2011 Feb; 86(1):97-116.*

- Crozier's paradox revisited: maintenance of genetic recognition systems by disassortative mating[BMC Evolutionary Biology. ]
*Holman L, van Zweden JS, Linksvayer TA, d’Ettorre P.**BMC Evolutionary Biology. 13211* - The Demographic Benefits of Belligerence and Bravery: Defeated Group Repopulation or Victorious Group Size Expansion?[PLoS ONE. ]
*Lehmann L.**PLoS ONE. 6(7)e21437*

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